1. Triangle Inequalities
Boardworks High School Geometry (Common Core)
Triangle Inequalities
Triangle Inequalities
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2. Boardworks High School Geometry (Common Core) Triangle Inequalities
Information
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3. Exploring triangle side lengths
Boardworks High School Geometry (Common Core)
Triangle Inequalities
Exploring triangle side lengths
Teacher notes
Use this activity to explore the triangle inequality before introducing the theorem.
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4. Triangle inequality theorem
Boardworks High School Geometry (Common Core)
Triangle Inequalities
Triangle inequality theorem
Triangle inequality theorem: The sum of any two sides of a triangle is greater than the third side.
AB + BC > AC AB + AC > BC BC + AC > AB
Two sides of a triangle have length AB = 4 and AC = 8. What can you say about the length of the third side, BC? C
By the triangle inequality theorem, the length of the third side must be strictly less than the sum of the two other sides: B
BC < AB + AC = = 12
BC < 12 A
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5. Testing the inequality theorem (1)
Boardworks High School Geometry (Common Core)
Triangle Inequalities
Testing the inequality theorem (1)
Teacher notes
Have students follow along with their rulers and compasses.
Remind students that when constructing a triangle, the longest side is always opposite the largest angle and the shortest side is opposite the smallest angle.
Mathematical practices
2) Reason abstractly and quantitatively.
Students should know from the triangle inequality theorem that the sum of the lengths of any two sides my best greater than the length of the other side. This activity gives them the opportunity to explore this concept quantitatively.
5) Use tools strategically.
Students will need to use rulers and compasses to follow along.
6) Attend to precision.
Students must carefully measure and draw the lines.
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6. Testing the inequality theorem (2)
Boardworks High School Geometry (Common Core)
Triangle Inequalities
Testing the inequality theorem (2)
Teacher notes
Consider using this activity as a quiz for applying the triangle inequality theorem. By clicking on the length labels on the triangle or the green boxes below the triangle, the lengths and inequalities can be hidden for students to determine.
Mathematical practices
8) Make use of repeated reasoning.
Students should see that the triangle inequality can be applied in three different ways to the same triangle.
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7. Finding possible lengths
Boardworks High School Geometry (Common Core)
Triangle Inequalities
Finding possible lengths
The length of one side can be any value greater than the difference and less than the sum of the other sides:
triangle inequality:
a < b + c
a different triangle inequality:
b < a + c
b – c < a for b > c ⇒
combining inequalities:
b – c < a < b + c
If b = 8 cm and c = 6 cm, what is the range of possible lengths for a? b
b – c < a < b + c
Teacher notes
Students should understand that the inequality statement above (b – c < a < b + c) is an example and not a generalization. In another triangle, b could be shorter than c, so if students subtracted c from b, the value would be negative.
Emphasize the generalization explained at the top of the slide: (difference of two sides) < (possible value for third side) < (sum of two sides)
Mathematical practices
1) Make sense of problems and persevere in solving them.
Students will need to understand that they are trying to find a range of value and so will need to apply the triangle inequality twice. They will also need to understand that this is only true when b>c.
2) Reason abstractly and quantitatively.
Here, students need to manipulate the inequality using symbols and apply it to given values. a
substituting:
8 – 6 < a < 8 + 6
2 < a < 14 c
a must measure between 2 cm and 14 cm.
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